It may thud in at a colossal 957 pages, but Bill Clinton's autobiography still feels incomplete. One closes this fattest of tomes with a sneaking suspicion that the saga is far from over. Its final words should read: To be continued.

That is not a prediction that Clinton is about to lead a one-man campaign to rewrite the constitution, enabling him to run again for the third term he surely wanted - and which, polls indicate, he would have won. No, it won't be Bill Clinton who seeks the White House again. Instead, it could well be a member of the Clinton family - almost certainly running against a representative of the clan that has become its arch rival: the Bushes.

It won't happen this time, but look ahead. If the Democrats lose this year and George Bush wins a second term, the 2008 battle will be wide open - with no incumbent on either side. Judged from here, the likeliest shape of that duel will be today's junior senator from New York v the current governor of Florida: Hillary Clinton v Jeb Bush.

Listen to the words of Bill Clinton himself. When Guardian editor Alan Rusbridger and I interviewed the former president last week, we threw in a final question about Hillary's presidential aspirations - confidently expecting it to be brushed aside with a politic insistence that the only name on the ballot is John Kerry and he is the candidate Democrats should focus on and support.

Clinton did a bit of that, declaring that politics is about the now and that Kerry "would be a good president if he won". But once that disclaimer was out of the way, he went on to speak - no, gush - about Hillary's many qualifications for the top job. He noted the parallels between her decision not to run this year and his own decision to sit out 1988 - a move that paid off when Michael Dukakis was defeated, paving the way for Clinton to run and beat the first President Bush four years later. The ex-president did not expect 1988 to be repeated this time, he said, guessing that Kerry has "a slightly better than 50-50 chance to win". But he had made his point. Equally telling, while he waxed generous and at length about Hillary's skills, he offered no such warm testimonial for Kerry.

If Hillary is gearing up for a run in 2008, so surely is the current president's brother, Jeb. When George W was still in his fratboy, party animal days - an adolescence that lasted until he was 40 - Jeb was being groomed to take over the family business. Steady and reliable, he would surely follow the line begun by his grandfather, Senator Prescott Bush, and continued by his father, George Bush senior. The plan only derailed in 1994 when Dubya won the governorship of Texas while Jeb lost in Florida.

So there is no shortage of appetite on either side of the dynastic divide. But would Hillary and Jeb be chosen by their respective parties? Hillary has won raves in the Senate and is regarded, along with her husband, as one of the very few Democratic stars. She only has to turn up to electrify a party audience, setting records for fundraising. A survey last October, before the primary season was under way, found Hillary was the first choice of 43% of Democratic voters; all the official candidates struggled to make double figures.

Jeb Bush does not yet have that kind of appeal: he remains, necessarily, in his brother's shadow. But he has one of the most precious commodities in US politics: name recognition. That factor propelled George W to success in 2000, enabling him to vault over his better-qualified rivals; there is no reason why the magic could not work again for Jeb.

But there is a larger reason why a Clinton-Bush match-up feels likely, if not inevitable. For these two dynasties have come to embody the two contrasting cultures of American life, the two worlds captured in the demographers' phrase: 50-50 America.

That political divide was exposed most clearly in the photo-finish presidential election of 2000. But it goes deeper. Look at the American bestseller lists, Michael Moore and Al Franken confirming the prejudices of the left, Ann Coulter and Bill O'Reilly feeding the biases of the right. Hear how the two sides insult each other: the right are Stupid White Men, the left are guilty of Treason.

Switch on the wireless: Rush Limbaugh for conservatives, National Public Radio for liberals. The TV: Fox on the right, PBS on the (perceived) left. New research by demographer James Gimpel shows that even the towns and neighbourhoods of the US are coming to look this way, as Americans engage in voluntary political segregation, choosing to live only with like-minded folk. The result is a "patchwork nation", reinforcing a map already painted in clear shades of red and blue. In Democrat blue are the west coast, rust belt and New England. In Republican red are the south, the plains states and the Rocky Mountain west.

What it amounts to is a divided nation, if not two nations living in one country. One watches Queer Eye for the Straight Guy and Will & Grace, the other wants to amend the constitution to outlaw gay marriage. To summarise even more crudely: one nation backs abortion rights, wants gun control, believes in the United Nations and lives in the cities and on the coasts; the other hopes to ban abortion, owns guns, believes America should do what it wants in the world and lives in the south and in the countryside.

The first tribe is now identified with the Democrats, the second with the Republicans. But the more visceral association is with people rather than parties. The first group has become embodied - at least in the eyes of their enemies - by the Clintons, while the second is personified by the Bushes. Which explains why both Bill Clinton and George W have become more than just disliked politicians, but hugely divisive hate figures. They personify more than a political programme, but an entire set of values - one that is despised by the other side.

Liberals are desperate to oust Bush, who they see as the living incarnation of the scolding, moralistic, gun-toting, Bible-bashing, ignorant right. Conservatives will do anything to stop the Clintons, who represent the dope-smoking, marriage-diluting, foreigner-loving, limp-wristed counterculture that they reckon has brought America close to ruin.

So it seems almost fated, that this battle will resolve itself into a simple, dynastic clash of Bushes against Clintons, the Montagues and Capulets of modern American life. After Hillary v Jeb, the battle will surely be carried to the next generation. Chelsea Clinton is an accomplished performer who may have inherited the best of her parents - the warmth of her father, the self-discipline of her mother; George P Bush, Jeb's son, is handsome, articulate and Hispanic - able to reach a community that could soon decide US elections. If the clash of dynasties does not come in 2008, watch out for November 2024. This is a family feud - and it's not over yet.